---
title: "Braced column 450x350, 4.0 m height"
subject: "Design of RCC Structure"
chapter: 10
key: "semester_7/design_of_rcc_structure:braced-column-450x350-4-0-m"
source: https://www.bctengineer.com/q/semester_7/design_of_rcc_structure/braced-column-450x350-4-0-m
site: BCT Engineering
---

# Braced column 450x350, 4.0 m height

Design of RCC Structure · Chapter 10: Design of compression members: Columns

- Asked in: 2080 Bhadra, 12 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/design_of_rcc_structure/braced-column-450x350-4-0-m
- Whole chapter: https://www.bctengineer.com/md/semester_7/design_of_rcc_structure/chapter-10

## Question

Design a braced rectangular RC column having clear height of 4.0 m, with cross sectional dimension of 450 mm x 350 mm subjected to design axial load of 700 kN, design bending moments of 100 kN-m about major axis and 80 kN-m about minor axis. The column is held effectively at both ends and restrained against rotation at one end. Consider M25 concrete and Fe415 steel.

## Short answer

Provide 12-16 mm bars (1.53%) and 8 mm ties @ 250 mm c/c. The biaxial interaction ratio is 0.992 ≤ 1.0, satisfying IS 456 requirements.

## Answer

#### Data and assumptions
- Column 350 mm (width $b$) × 450 mm (depth $D$); $P_u = 700$ kN; $f_{ck} = 25$ MPa, $f_y = 415$ MPa. Clear cover 40 mm (IS 456 cl. 26.4.2.1), 8 mm ties.
- Moments: about major axis (x-x) $M_{ux}$ = 100 kN·m; about minor axis (y-y) $M_{uy}$ = 80 kN·m.
- Held in position at both ends and restrained against rotation at one end: $l_e = 0.8L = 0.8 \times 4.0 = 3.2$ m (IS 456 Table 28).
- Sizes are given as $b \times D$ with $D \ge b$; moments $M_{ux}$ act about the major axis (depth $D$) and $M_{uy}$ about the minor axis (depth $b$).

#### Slenderness
| Axis | $l_e$ (mm) | Dimension (mm) | $l_e/$dim | Type |
|---|---|---|---|---|
| x-x (about major) | 3200 | 450 | 7.11 | short |
| y-y (about minor) | 3200 | 350 | 9.14 | short |
Both ratios are below 12, so the column is **short**; no additional moment is required.

#### Minimum eccentricity (cl. 25.4)
$$
e_{min} = \max\left(\frac{L}{500} + \frac{D}{30},\ 20\right)
$$
- About x-x: $e_x = 4000/500 + 450/30 = 23.0$ mm (min. 20 mm); adopt 23.0 mm; $M_{ux,min} = P_u e = 16.1$ kN·m
- About y-y: $e_y = 4000/500 + 350/30 = 19.7$ mm (min. 20 mm); adopt 20.0 mm; $M_{uy,min} = 14.0$ kN·m

#### Design moments
- Design moments: $M_{ux} = 100.0$ kN·m, $M_{uy} = 80.0$ kN·m (not less than the minimum-eccentricity values).

#### Longitudinal reinforcement
$\dfrac{P_u}{f_{ck} b D} = \dfrac{700000}{25 \times 350 \times 450} = 0.178$
Bars are chosen so that $p$ lies between 0.8% (minimum) and 4% (practical maximum; IS 456 cl. 26.5.3.1 allows 6%). The uniaxial capacities $M_{ux1}$, $M_{uy1}$ at $P_u$ are read from the SP-16 interaction charts for rectangular sections; here they are computed by strain compatibility with the IS 456 stress block and the design stress-strain curve of steel (same basis as the charts), with effective cover $d' = 40 + 8 + \phi/2$.
| Trial | $p$ (%) | $d'/D$ | $M_{ux1}$ | $M_{uy1}$ | $P_u/P_{uz}$ | $\alpha_n$ | $(M_{ux}/M_{ux1})^{\alpha_n}+(M_{uy}/M_{uy1})^{\alpha_n}$ |
|---|---|---|---|---|---|---|---|
| 8-16 φ | 1.02 | 0.124 | 163.3 | 120.9 | 0.311 | 1.18 | 1.173 ✗ |
| 10-16 φ | 1.28 | 0.124 | 171.1 | 137.0 | 0.295 | 1.16 | 1.073 ✗ |
| 12-16 φ | 1.53 | 0.124 | 194.4 | 142.0 | 0.280 | 1.13 | 0.992 ✓ |

#### Check
$P_{uz} = 0.45 f_{ck} A_c + 0.75 f_y A_{sc} = 0.45 \times 25 \times 155087 + 0.75 \times 415 \times 2413 = 2496$ kN
$\dfrac{P_u}{P_{uz}} = \dfrac{700}{2496} = 0.280$, so $\alpha_n = 1.13$ (IS 456 cl. 39.6; linear between 1.0 at 0.2 and 2.0 at 0.8).
$$
\left(\frac{M_{ux}}{M_{ux1}}\right)^{\alpha_n} + \left(\frac{M_{uy}}{M_{uy1}}\right)^{\alpha_n} = \left(\frac{100.0}{194.4}\right)^{1.13} + \left(\frac{80.0}{142.0}\right)^{1.13} = 0.992 \le 1.0
$$
The interaction condition is satisfied; the section is **safe**.

Provide **12-16 mm** bars ($A_{sc} = 2413$ mm², $p = 1.53\%$), within 0.8% to 4%.

#### Lateral ties
- Diameter $\ge \max(\phi_{max}/4,\ 6) = \max(16/4, 6) = 6$ mm → provide 8 mm.
- Pitch $\le \min$(least lateral dimension, $16\phi_{min}$, 300 mm) $= \min(350, 256, 300) = 256$ mm → provide 8 mm ties @ 250 mm c/c.

#### Detailing
```text
CROSS-SECTION 350 x 450
    +-------------------------+
    |   o     o     o     o   |
    |                         |
    |                         |
    |   o                 o   |
    |                         |
    |                         |
    |                         |
    |   o                 o   |
    |                         |
    |                         |
    |   o     o     o     o   |
    +-------------------------+
   o = 16 mm bars, ties 8 mm @ 250 c/c
```

**Answer:** 350 × 450 mm column with 12-16 mm longitudinal bars ($p = 1.53\%$) and 8 mm ties @ 250 mm c/c.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. For a column held in position at both ends and restrained against rotation at one end, what is le?
   A. 0.65L
   B. 0.8L = 3.2 m
   C. 1.0L
   D. 1.2L
   Answer: B. IS 456 Table 28 gives le = 0.8L for this condition, so le = 0.8 x 4.0 = 3.2 m. (Key point: Effective length)

2. Why were 8-16φ and 10-16φ rejected and 12-16φ adopted?
   A. They violate the minimum steel limit
   B. Their interaction sums were 1.173 and 1.073 (> 1), while 12-16φ gives 0.992
   C. They exceed 4% steel
   D. They are not symmetric
   Answer: B. The section is safe only if the biaxial interaction sum is <= 1. Only 12-16φ achieves this. (Key point: Why trial 3 adopted)

3. With Pu/Puz = 0.280, what is αn?
   A. About 1.13
   B. About 1.50
   C. 2.0
   D. 1.0
   Answer: A. αn = 1 + (0.280 - 0.2)/(0.8 - 0.2) = 1.13. (Key point: Interaction exponent)

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